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Black hole complementarity posits that the interior of a black hole is not independent from its Hawking radiation. This leads to an apparent violation of causality: the interior can be acausally affected by operators acting solely on the radiation. We argue that this perspective is misleading and that the black hole interior must be viewed as existing in the causal past of the Hawking radiation, despite the fact that they are spacelike separated in the semiclassical description. Consequently, no operation on the Hawking radiation -- no matter how complex -- can affect the experience of an infalling observer. The black hole interior and the radiation only appear spacelike separated in the semiclassical description because an infalling observer's ability to access complex information is limited; the chaotic dynamics on the horizon, as viewed from the exterior, then converts any effect caused by such an observer to information in the Hawking radiation which cannot be accessed at the semiclassical level. We arrive at the picture described above by considering a unitary exterior description in which the flow of information is strictly causal, which we extend to apply throughout the entire history of black hole evolution, including its formation. This description uses the stretched event horizon as an inner edge of spacetime, on which the information inside is holographically encoded. We argue that the global spacetime picture arises from coarse-graining over black hole microstates, and discuss its relationship with the exterior description., Comment: 37 pages, 12 figures

We describe two different, but equivalent semiclassical views of black hole physics in which the equivalence principle and unitarity are both accommodated. In one, unitarity is built-in, while the black hole interior emerges only effectively as a collective phenomenon involving horizon (and possibly other) degrees of freedom. In the other, more widely studied approach, the existence of the interior is manifest, while the unitarity of the underlying dynamics can be captured only indirectly by incorporating certain nonperturbative effects of gravity. These two pictures correspond to a distant description and the description based on entanglement islands/replica wormholes, respectively. We also present a holographic description of de Sitter spacetime based on the former approach, in which the holographic theory is located on the stretched horizon of a static patch. We argue that the existence of these two approaches is rooted in the two formulations of quantum mechanics: the canonical and path integral formalisms., Comment: 103 pages, 23 figures

We examine non-relativistic holographic RG flows by working with Einstein-Maxwell-scalar theories which support geometries that break Lorentz invariance at some energy scale. We adopt the superpotential formalism, which helps us characterize the radial flow in this setup and bring to light a number of generic features. In particular, we identify several quantities that behave monotonically under RG flow. As an example, we show that the index of refraction is generically monotonic. We also construct a combination of the superpotentials that flows monotonically in Einstein-scalar theories supporting non-relativistic solutions, and which reduces to the known c-function in the relativistic limit. Interestingly, such quantity also exhibits monotonicity in a variety of black hole solutions to the full Einstein-Maxwell-scalar theory, hinting at a deeper structure. Finally, we comment on the breakdown of such monotonicity conditions and on the relation to a candidate c-function obtained previously from entanglement entropy., Comment: 34 pages, 8 figures. Minor comments added and references added